There is no unique way to couple prior and posterior distributions in
a Bayesian inference problem.
In the context of filtering,
we need couplings that can be estimated efficiently in high dimensions from finitely many prior samples.
In Sections  and
,
we discuss specific choices for these couplings, which lead to novel
nonlinear filtering algorithms.
Before that, however, we need to discuss the
properties of an important transport for our analysis,
the Knothe–Rosenblatt rearrangement [1]}.
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